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\hyphenation{op-tical net-works semi-conduc-tor u-san-do u-sa-mos}

\begin{document}

\title{		Redes de HopField}
\author{	Jos\'e Liendro~\IEEEmembership{(Legajo I.T.B.A.: 48.240 ),}\\
        	Damian Modernell~\IEEEmembership{(Legajo I.T.B.A.: 45.002),}\\
			Cristian Prieto~\IEEEmembership{(Legajo I.T.B.A.: 47.573)}
}

\IEEEcompsoctitleabstractindextext{%
\begin{abstract}
%\boldmath
 En este informe se desarrolla la implementaci\'on de una red de Hopfield, y se analizan resultados obtenidos luego de varios
conjuntos de pruebas. En la primera secci\'on se detalla brevemente el problema en cuestion, luego se da un
desarrollo y explicaci\'on de las pruebas realizadas, finalmente se obtienen las conclusiones sobre la utilizaci\'on de redes de este tipo.
\end{abstract}


\begin{IEEEkeywords}
Red de Hopfiled, atractores, memoria asociativa, regla de actualizaci\'on asincr\'onica.
\end{IEEEkeywords}}
\maketitle

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\section{Introducci\'on}
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\IEEEPARstart{E}{}l objetivo de este informe es  analizar el comportamiento de una red de Hopfield, que act\'ue como memoria
asociativa direccionable. En la secci\'on 2 se describe la metodolog\'ia de pruebas realizadas, en la secci\'on 3 se describen los
resultados obtenidos, y posteriormente las conclusiones alcanzadas.
% You must have at least 2 lines in the paragraph with the drop letter
% (should never be an issue)

\section{Desarrollo de la red}
Para analizar el comportamiento de la red, se toman como patrones, im\'agenes en blanco y negro que la red debe memorizar.
El objetivo es analizar qu\'e tan bien la red logra memorizar dos o m\'as patrones simult\'aneamente, y luego, una vez obtenida la red, determinar, pas\'andole
distintos patrones de entrada que tan bien logra reproducirlos.\\
Las im\'agenes se identificar\'an con un nombre que se puede encontrar en la secci\'on anexa.\\
Las pruebas son las siguientes:
\section{Pruebas realizadas}

\begin{enumerate}
\item  Memorizaci\'on de las im\'agenes A, F, H y prueba de la evoluci\'on a estado estable de cada una por separado.
\item  Idem prueba 1 con la imagen A_mod en lugar de A la cual sit\'ua a la letra 'A' en en una posici\'on distinta, no superpuesta con la H.
\item  Idem prueba 1 a\~{n}adiendo dos im\'agenes para reforzar la memorizaci\'on.
\item  Idem prueba 1 con prueba de la evoluci\'on a estado estable de partes incompletas de esas im\'agenes.
\item  Idem prueba 1 con prueba de la evoluci\'on a estado estable de patrones(im\'agenes) que no pertenecen al conjunto memorizado.
\item  Idem prueba 1 con prueba de la evoluci\'on a estado estable de las im\'agenes memorizadas con ruido.
\item  Idem prueba 1 con prueba de la evoluci\'on a estado estable de las im\'agenes memorizadas con colores invertidos.
\item  Memorizaci\'on de las im\'agenes camara, joystick y gameboy y evoluci\'on a estado estable de cada una.
\item  Idem prueba 8 con prueba de la evoluci\'on a estado estable de las im\'agenes memorizadas con ruido.
\item  Memorizaci\'on de dos im\'agenes ( circulo1 y circulo2 en posiciones sin intersecci\'on de puntos ) y evoluci\'on a estado estable de ambos patrones por separado.
\end{enumerate}

\section{Resultados}


\subsection{Prueba 1}

\begin{table}[H]
%% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
  \label{fig:nonfloat}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
\caption{}
\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{c c c c}
\hline
\hline
Memorizados & patron  & evoluci\'on\\
\hline
 (A, F, H) & A & evoluciona a H\\
 (A, F, H) & F & evoluciona a F \\
 (A, F, H) & H & evoluciona a H \\
\hline
\hline
\end{tabular}
\end{table}
Se puede observar que como la imagen A es id\'entica a la H excepto por los bits de la parte de arriba de la letra 'A'.
Por lo tanto la red toma a la imagen A como la H con cierto n\'umero de bits de ruido.

\subsection{Prueba2}

\begin{table}[H]
%% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
  \label{fig:nonfloat}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
\caption{}
\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{c c c c}
\hline
\hline
Memorizados & patron  & evoluci\'on\\
\hline
 (A, F, H) & A & evoluciona a H\\
 (A, F, H) & F & evoluciona a F \\
 (A, F, H) & H & evoluciona a H \\
\hline
\hline
\end{tabular}
\end{table}

Se puede observar que

\subsection{Prueba 2}

\begin{table}[H]
%% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
  \label{fig:nonfloat}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
\caption{}
\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{c c c c}
\hline
\hline
Memorizados & patron  & evoluci\'on\\
\hline
 (A_mod, F, H) & A_mod & evoluciona a A_mod\\
 (A_mod, F, H) & F & evoluciona a F \\
 (A_mod, F, H) & H & evoluciona a H \\
\hline
\hline
\end{tabular}
\end{table}
Se observa que con la imagen A_mod se logra que a letra 'A' evolcione a 'A' y no a 'H'


\subsection{Prueba 3}

\begin{table}[H]
%% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
  \label{fig:nonfloat}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
\caption{}
\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{c c c c}
\hline
\hline
Memorizados & patron  & evoluci\'on\\
\hline
 (A, F, H, line3, line4) & A & evoluciona a A_bis\\
 (A, F, H, line3, line4) & F & evoluciona F_bis \\
 (A, F, H, line3, line4) & H & Evoluciona a H \\
\hline
\hline
\end{tabular}
\end{table}
En este caso la im\'agen line4 refuerza la memorizaci\'on de A. Esto trajo como resultado que la imagen F ya no evolucionara a s\'i misma.
Es por eso que se agrega la imagen line3 que hace que la imagen F evolucione a F_bis bastante cercana a la original F.


\subsection{Prueba 4}

\begin{table}[H]
%% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
  \label{fig:nonfloat}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
\caption{}
\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{c c c c}
\hline
\hline
Memorizados & patron  & evoluci\'on\\
\hline
 (A, F, H) & parte peque\~{n}a de A & evoluci\'on irreconocible de A\\
 (A, F, H) & parte mediana de A & evoluci\'on irreconocible de A\\
 (A, F, H) & parte grande de A & evoluci\'on a A\\
 (A, F, H & parte peque\~{n}a de F & evoluci\'on a F \\
 (A, F, H) & parte peque\~{n}a de H & Evoluciona a H \\
\hline
\hline
\end{tabular}
\end{table}


\subsection{Prueba 5}

\begin{table}[H]
%% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
  \label{fig:nonfloat}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
\caption{}
\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{c c c c}
\hline
\hline
Memorizados & patron  & evoluci\'on\\
\hline
 (A, F, H) & mouse & evoluci\'on a F\\
 (A, F, H) & c\'amara & evoluci\'on a H\\
 (A, F, H) &pent\'agono & evoluci\'on a H\\
 (A, F, H & circulo & evoluci\'on a F \\
\hline
\hline
\end{tabular}
\end{table}

\subsection{Prueba 6}
\begin{table}[H]
%% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
  \label{fig:nonfloat}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
\caption{}
\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{c c c c}
\hline
\hline
Memorizados & patron  & ruido &  evoluci\'on\\
\hline
 (A, F, H) & A  & 10&  evoluci\'on a H\\
 (A, F, H) & F & 10 & evoluci\'on a F\\
 (A, F, H) & F & 15 & evoluci\'on a F\\
 (A, F, H) & F & 20 & evoluci\'on a H\\
 (A, F, H) & H & 15 & evoluci\'on a H\\
 (A, F, H) & H & 20 & evoluci\'on a H\\
 (A, F, H) & H & 25 & evoluci\'on a H\\
 (A, F, H) & H & 30 & evoluci\'on a H\\
 (A, F, H) & H & 64 & evoluci\'on a H\\
\hline
\hline
\end{tabular}
\end{table}


\subsection{Prueba 7}

\begin{table}[H]
%% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
  \label{fig:nonfloat}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
\caption{}
\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{c c c c}
\hline
\hline
Memorizados & patron  &   evoluci\'on\\
\hline
 (A, F, H) & A invertida &  evoluci\'on a H invertida\\
 (A, F, H) & F invertida &  evoluci\'on a F invertida\\
 (A, F, H) & H invertida &  evoluci\'on a H invertida\\
\hline
\hline
\end{tabular}
\end{table}


\subsection{Prueba 8}

\begin{table}[H]
%% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
  \label{fig:nonfloat}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
\caption{}
\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{c c c c}
\hline
\hline
Memorizados & patron  & evoluci\'on\\
\hline
 (Joystick, C\'amara, Gameboy) & Joystick &  evoluci\'on a Joystick\\
 (Joystick, C\'amara, Gameboy) & Gameboy &  evoluci\'on a Gameboy\\
 (Joystick, C\'amara, Gameboy) & C\'amara &  evoluci\'on a C\'amara\\
\hline
\hline
\end{tabular}
\end{table}


\subsection{Prueba 9}

\begin{table}[H]
%% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
  \label{fig:nonfloat}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
\caption{}
\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{c c c c}
\hline
\hline
Memorizados & patron  & ruido &  evoluci\'on\\
\hline
 (Joystick, C\'amara, Gameboy) & Gameboy & 1 &evoluci\'on irreconocible \\
 (Joystick, C\'amara, Gameboy) & Gameboy & 2 &evoluci\'on irreconocible \\
 (Joystick, C\'amara, Gameboy) & Gameboy & 5 &evoluci\'on irreconocible \\
 (Joystick, C\'amara, Gameboy) & Gameboy & 10 &evoluci\'on irreconocible \\
 (Joystick, C\'amara, Gameboy) & Joystick & 2 &evoluci\'on a Joystick\\
 (Joystick, C\'amara, Gameboy) & Joystick & 5 &evoluci\'on irreconocible \\
 (Joystick, C\'amara, Gameboy) & Joystick & 10 &evoluci\'on irreconocible\\
 (Joystick, C\'amara, Gameboy) & C\'amara & 10 &evoluci\'on a c\'amara\\
 (Joystick, C\'amara, Gameboy) & C\'amara & 20 &evoluci\'on irreconocible\\
\hline
\hline
\end{tabular}
\end{table}

\subsection{Prueba 10}

\begin{table}[H]
%% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
  \label{fig:nonfloat}

% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
\caption{}
\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{c c c c}
\hline
\hline
Memorizados & patron  & evoluci\'on\\
\hline
 Dos im\'agenes(c\'irculo1, c\'irculo2) & c\'irculo1 & evoluci\'on correcta a c\'irculo1\\
 Dos im\'agenes(c\'irculo1, c\'irculo2) & c\'irculo2 & evoluci\'on correcta a c\'irculo2\\
\hline
\hline
\end{tabular}
\end{table}



%\begin{figure}[H]
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%\centering
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% \DeclareGraphicsExtensions.
%\caption{Aprendizaje utilizando la funcion tangente hiperb\'olica, variando el coeficiente de aprendizaje }
%\end{center}
%\end{figure}


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\section{Conclusi\'on}

%Con el modelo que propusimos de la batalla de Iwo Jima, expresado por las ecuaciones diferenciales 8 y 9, 
%nos aproximamos considerablemente a los valores reales. Sin embargo, el modelo puede ser mejorado agregando a las ecuaciones factores que 
%pueden influir en la cantidad de bajas de cada ej\'ercito, como por ejemplo estimar una tasa de p\'erdidas operativas
%que puede depender de la calidad m\'edica, la alimentaci\'on, o las deserciones de ambas fuerzas.







% if have a single appendix:
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% or
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%\appendices
%\section{Proof of the First Zonklar Equation}
%Appendix one text goes here.

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 % \section*{Acknowledgments}
%\else
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%The authors would like to thank...


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% references section

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\bibitem{IEEEhowto:kopka}
 Hertz, John. \emph{Introduction to the theory of neural computation}


\end{thebibliography}




\end{document}


